Powers of Ten

or exponential notation on a
scientific calculator.

In science, we deal with numbers that are sometimes extremely large or
extremely small. It helps to have a calculator around when doing math with
such large numbers but you have to know how to use one.

There are
602,000,000,000,000,000,000,000 molecules in 18 grams of water. A shorter
way of writing the same number is by using exponential notation to show
all those zeros as a number to the power of ten:

6.02 x 1023 is the shorter way of representing all
those molecules. Such a number can be read "Six point zero two times
ten to the twenty third."

A small number such as 0.0000000057 can be written as 5.7 x 10-9.
Such a number can be read "Five point seven times ten to the minus
nine."

How do you convert from one way of writing to another?

Well, it's all in how you move the decimal point. Writing 104,
for example, means that you are writing a one and then moving the decimal
point four places. Since the exponent (the 4) is a positive number, we
move the decimal place to the right. The animation below shows how
it is done:

Notice that the animation starts by reminding us that when
we write "1", we are really writing "1."
but we usually do not put the decimal point. At the end of the
animation, the decimal point is in its new place, but again, we do not
write it. Notice, also that there is a total of four zeroes.
This is one way of checking if you did it correctly. (104,
gives 4zeroes).

Now let's try the opposite. What if the exponent is
a negative number? With the number 10-4, the
technique starts out the same - we write a number 1, remembering that
there is a decimal point after it. But then instead of moving the
decimal point to the right, we move it to the left.

To remember which way to move (left or right), remember that negative
exponents (like -4 in the example) represent very small numbers - numbers
that are less than one. As a result, they will always start with
zero. Positive exponents (like 4 in the example), give numbers
bigger than one.

On scientific calculators, it is sometimes a bit of a trick to type
these numbers in.

To type 104, one way of doing it is to type
ten:

Then use the xy button
and type 4

The result should be ten thousand:

But there is a quicker way, using a special
button which has been designed especially for exponential
notation.

To type 10-4, type a one* and then hit the x
10x button:
Can't find this button on your calculator? See the next
box.

You should get this:

Now type a four

This number means 1 x 104 which
is the same as 104.

If your calculator works like this, you can skip
the next box.

*Why did we type 1 and not 10?

Let's see what happens when we try ten.

If you press =, you get

That's because the first screen (10.04)
means 10 x 104. This number is the same
a writing 105 which is 100000.

Other
calculators have "EXP"
or "EE" buttons on them for scientific notation. Certain calculators have a button
which can be pushed directly, like the one on the right, others have this function available as a
two-key operation.

On some calculators, you have to push the 2nd command (or Shift button)
first. Here is an example:

+

=

"x 10x"

On these the expression "x 10x" is
replaced by the letter "E".

Let's take an example such as 6.23 x 1023
times 4.11.

Type in 6.23 then the EE button, then 23. After that,
type times 4.11 and Enter. Here is what it would look like on the
calculator's screen:

Notice how the expression "x 10"
has been replaced by the letter "E".
This is to save space on the small screen and to avoid confusion with
multiplication operations.

The letter "E"
is placed there to remind us that this number is written in "Exponential
notation".

For a calculator, 6.23E23
equals 6.23 x 1023.

The answer above, 2.56E24
would be written on your paper 2.56 x 1024.

Let's stay with calculators
which have only one line on
the screen which do not put the letter "E"
at all but rather put the exponent up in the right hand corner of the
screen in a smaller format. A number such as 5.98 x 10-6
looks like this:

How do you type in numbers like this?
Let's try 0.0000371, which is written in scientific notation as 3.71 x 10-5.

First, type in three point seven one:

Next, push the x
10x button to get into the scientific notation
mode:

To make the exponent negative, push the +/- button:
which on some calculators is a negative sign in parentheses:
(-).

Important: do NOT use the minus sign, this is a totally
different operation!

The screen should now look like this:

Type in five to get the final number:

This is how your calculator says 3.71 x 10-5. If
you get an answer like this from a calculation, be sure you
write it the way humans do (3.71 x 10-5) and
not the way your calculator does (3.71-05 or 3.71E-05).
Written on a piece of paper, the number 3.71-05
is equivalent to 0.00142 and not what we want, 0.0000371.
It's obvious that these are completely different numbers!

A more simple example is that 5 x 102 is not the
same as 52. The first equals 500 and the second
equals 25.